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On the asymptoticity aspect of Hyers-Ulam stability of mappings. (English) Zbl 0894.39012
Let \(E_1\) be a real normed vector space and \(E_2\) a real Banach space, let \(\varepsilon>0\). Answering a problem posed by S. M. Ulam, in 1941 D. H. Hyers [Proc. Natl. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.26403)] proved that if \(f:E_1\to E_2\) satisfies \(\| f(x+y)-f(x)-f(y)\| \leq \varepsilon\) for all \(x,y \in E_1\), then there exists a unique additive mapping \(T:E_1\to E_2\) such that \(\| f(x)-T(x)\| \leq\varepsilon\) for all \(x \in E_1\).
The assumptions were weakened by Th. M. Rassias [Proc. Am. Math. Soc. 72, 297-300 (1978; Zbl 0398.47040)], who proved that if \(f\) satisfies, for some \(0< p < 1\), \(\| f(x+y)-f(x)-f(y)\| \leq \varepsilon (\| x\| ^p+\| y\| ^p)\) for all \(x,y \in E_1\), then there is a unique additive mapping \(T:E_1\to E_2\) with \(\| f(x)-T(x)\| \leq \varepsilon \beta(p) \| x\| ^p\) for all \(x\in E_1\), where \(\beta(p)=2/(2-2^p)\).
The present paper generalizes this result further. The authors show that if \(f\) satisfies \(\| f(x+y)-f(x)-f(y)\| \leq\varepsilon (\| x\| ^p+\| y\| ^p)\) whenever \(\| x\| ^p+\| y\| ^p>M^p\) for some \(M>0\), then there is an additive mapping \(T:E_1 \to E_2\) with \(\| f(x)-T(x)\| \leq \varepsilon \beta(p) \| x\| ^p\) for all \(\| x\| >M/2^{1/p}\).

MSC:
39B72 Systems of functional equations and inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
39B52 Functional equations for functions with more general domains and/or ranges
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